Hi Günther,
On 07 Jan 2009, at 22:47, Günther Greindl wrote:
>
> thanks for your comments, I interleave my response.
>
>>> showed a glimpse of the vastness of the UD. And, I agree, _in the
>>> limit_
>>> there will be an infinite number of histories. So, as we have to
>>> also
>>> take into account infinite delay, we must take this limit into
>>> account
>>> and have infinite histories going through a "state" (do I
>>> understand you
>>> correctly?).
>>
>> I guess you were meaning that we have to take into account an
>> infinity
>> of arbitrary long (but finite) delays. OK.
>
> Hmm, if we have an infinity of arbitrary long but finite delays,
> then I
> can only see aleph_0 histories (because we never take the "step to
> infinity" - we can enumerate all histories.
>
> Only if we take the "step to infinity" (as in Cantor diagonalization,
> were we presuppose the complete listing of the reals and the diagonal
> does not fit "at infinity") would we get 2^aleph_0 histories - or am I
> missing something here?
Cantor's proof is a "reductio ad absurdo". It assumes there is a one
one correspondence, or bijection, between the positive integers and
the infinite sequence on {0, 1} say. Such correspondence could be
partially described by the diagram
1 ---- 10010111100 ...
2 ---- 01101001100 ...
3 ---- 11000100111 ...
4 ---- 11101111000 ...
5 ---- 10100110101 ...
6 ---- 00010111011 ...
7----
...
and Cantor get a contradiction from that. You assume the diagram is
indeed a piece of an existing bijection in Platonia, or known by God.
But if such a bijection exist, or if God can conceive that
correspondence, then there is a special sequence that God can conceive
too, and that indeed you can "bulld" from that diagram, indeed the
sequence
001110...
that you get by flipping the 0 and 1 along the diagonal of the matrix
appearing on the right in the diagram. That sequence, thus, exists in
Platonia, but definitely cannot belong to the list described above. If
it was in the list, there would be a number k
k --- 001110...
But by definition of the sequence, the kth decimal of that number k
will be the flip of itself, meaning that 0 = 1. OK?
The reasoning did not depend on the choice of any one one
correspondence, so that we know that for each correspondence there is
a corresponding anti-diagonal sequence, refuting the assertion that
correspondence could exhaust the set of all infinite binary sequences.
The set of binary sequence is thus not listable, not enumerable, not
countable.
You can visualise geometrically the contradictions for any candidate
correspondence by the intersection of the line defined by the
corresponding number k and the diagonal of the matrix describing the
correspondence. Note that the diagonal makes to contradiction
appearing always in a finite time.
I insist on this diagonal because it is the main tool of the AUDA. A
very similar diagonal shows the existence of enumerable but non
recursively enumerable set of numbers, which have some role in
"machine's theology" (or more quotes).
But then, recall the UD dovetails on the infinite computation, and
sometimes dovetails those infinite computation with the generation of
the binary sequences.
So you have to look at it, in the third person point of view as
computations which bifurcate (or differentiate by the rule Y = II),
and bifurcate again, and again, and again, OK?
And now, what you are missing. I think. It is the distinction between
third and first person point of view. As defined in the first and
second step of UDA (not the Theatetical one used in AUDA).
Looking at the generation of the UD, or dovetailing on all
computations, you can see the many computations being generated and
you can see them differentiate or bifurcating all the "time", where
here time is defined by the succession 1, 2, 3, 4, 5, ... itself. If
you universal base is two dimensional (like with the Conway Game of
Life) you can see the deployment as a static three dimensional conic
structure.
Everything there, is enumerable. At each UD step, everything is even
finite!
But things changes when you adopt the first person point of view, due
to the fact that the first person point of view cannot be aware of the
dovetailing delays, nor of the extreme multiplication and redundancy
of the computations. And if you are OK with, well, mainly here the
step 4, you see that the "intuitive" measure will have to be made on
the union of all computations going through the "current" state.
There is a continuum of such infinite computations, if only due to
that entangling of computation on the dovetailing on the reals and the
Y = II rule.
The third person probabilities for the *first* person point of views
have to bear on the fact that although the reals or the binary
sequence are not enumerable it is easy to write a simple program which
generates them all. This is not always well understood, but the trick
is very simple; just don't name them. In that way when just writing
(here "..." *is* a pure symbol)
0...
I have already begin the generation of a continuum of binary
"history": Indeed, all those beginning by 0. Then I write
1...
So I have begin the generation of the binary sequence beginning by 1.
As you see I am dovetailing (not universally though!).
Then i generate all possible extensions, which give me two time more
work.
First the possible continuation of the one beginning by 0.
00...
01...
Then the possible continuations of 1
10...
11...
Then:
000...
001...
010...
011...
100...
101...
110...
111...
Each time the work double, from two beginnings to four then eight,
then 16, then 32, then 64, etc. The diophantine exponential 2^x.
Now, if you interpret the 1 or 0 as results of a self-bifurcation in
the UDA, then by the unawareness of delays, the first person
indeterminacy of those "in front of a never stopping UD", where your
computations are dovetailed, in particular on the binary infinite
sequences, bears on set with cardinalities of continua, despite
mathematically the third person description does not leave the
enumerable.
I can understand this is "shocking", but not so more than with
Everett. It is not so astonishing when you think that those continua
described our first person ignorance and indeterminacies.
Coming back for a second to the third person points view, you can
contemplate many impressive infinite sequence in the Mandelbrot sets,
they converge either toward the "little mandelbrot", or toward deeper
bifurcating sequences. It is typical of the doubling scenario of
chaotic regime.
>
>
>> I will let you elaborate on this. But note that if my consciousness
>> "here and now" supervenes on "past activity",
>
> I will elaborate, but please give me time till February, before I will
> not be able to work on this.
Take your time.
>
>
>> then the comp substitution
>> level has to be very "low" indeed.
>
> Yes, very low, that was the idea.
>
>> You will also need a notion of "block
>> universe". The comp doctor will have to be able to manipulate
>> "time-lines".
>
> No, it is only that he will have to respect "relative embeddings" -
> scanning and reconsitution will only be correct regarding _this_
> universe and very similar universes, but not with regard to arbitrary
> computations in Platonia.
OK, the level is so low that "you" = the universe, not in the
solipsistic sense. You say yes to the doctor provided that he will
simulated the "whole universe" at some correct level of description.
This change nothing in the reasoning, and the laws of such universes
remains "reduced" or "reducible in principle" to the proportion of
"arbitrary histories", but perhaps no so arbitrary given that they
have to go through your current state.
The UD respect all the relativities, and you could be right, even with
only low levels (not necessarily the "bottom"). Also, I can sometimes
speculate that comp could predict there is no bottom.
>
>
>> Remember that even deep, in the sense of Bennett(*) ,
>> computer state, can be copied efficiently, so that when you say that
>> (*) Bennett, C. H. (1988). Logical Depth and Physical Complexity. In
>> Herken, R., editor, /The Universal Turing Machine A Half-Century
>> Survey/, pages 227-258. Oxford University Press.
>>
>
> Thanks for the reference, I will consider this...
>
>
>>> If we assume the whole universe to be an automaton, also it's
>>> inhabitants would be "mechanical" =effectively computable of
>>> course -
>>> but maybe they could then not be duplicated, because, when person
>>> A were
>>> trying to make a scan of the properties of person B, the universe
>>> as a
>>> whole would move into different states and make complementary
>>> observables - which _could_ be necessary for a duplication -
>>> unavailable.
>>
>>
>> OK. But your level has to be really at the bottom, not only below the
>> quantum level. I recall you that the no-cloning theorem does not
>> prevent
>> us to be quantum computer. Right: we cannot say yes to any doctor,
>> yet
>> UDA goes through because at the seventh step the "copy need" is
>> eliminated. We need only turing emulability, because quantum states,
>> although not copyable, are "preparable" (in the quantum "prepare"
>> sense)
>> in many exemplaries, and indeed the UD does doevetail on all quantum
>> computations.
>
> Agreed.
>
>> I think that your bottom really means: my brain is the whole of
>> reality.
>
> In the sense that the brain state depends on the whole of reality, and
> if my brain state (or anyone elses) changes then the whole universe
> transitions into a new state, yes, but not in the solipsitic form.
I respect the non computationalist (as far as he respects the
computationalist), and I respect the "low-level" computationalist too
of course. I try to runaway, but not always succeed, in front of
solipist, which in my opinion should be helped in asylum.
The real question is what will you think if you, "low-level"
computationalist father have a daughter falls in love with a high-
level computationalist?
It is a complex puzzle because although they share the same basic
theology they will have quite different theotechnologies.
>
>
>>> This may only work if consciousness supervenes not on "isolated"
>>> computations, but only if it is embedded in computations
>>> constructing
>>> whole universes - but then again, we can't exclude this a priori.
>>>
>>> And we would still have to consider many worlds, but these would
>>> then
>>> indeed be _worlds_ and not only OMs with incoming/outgoing
>>> histories (of
>>> course it would still seem this way from the concrete OM, but with
>>> greatly reduced danger of white rabbits) - the laws of physics
>>> would not
>>> emerge for OM's stranded in the UD deployment but for OM's embedded
>>> already in highly structured computational environments - we would
>>> only
>>> have to take into account duplications of OM's where also whole
>>> universes are duplicated.
>>
>>
>> Hmmm.... (I guess I use "OM" in a larger sense: those worlds remain
>> computable (assuming comp and "bottom-level") and, as such, are
>> generated by the UD). I guess I should not!
>
> Could you please clarify what exactly you mean with OM? Maybe this can
> clear up some misunderstandings?
OM are Nick Bostrom's subjective "observer moment". Basically,
momentaneous qualia of feeling to be "in space-history".
I use sometimes OM in that sense, although I tend to write 1-OM for it.
By 3-OM I mean either a computational state "of a brain or of a
universal machine "vehiculating" that experience, that quale"
Or, in probabilistic context, a 3-OM is identified with all its
occurence in the UD deployment. It is the many 3-OM, corresponding to
the same experience (qualitatively, and this exists in infinity in the
UD deployment, quantitatively, assuming comp).
>
>
>
>> Well, if the quantum laws are derived from comp, then the "platonic
>> histories" are manipulable in a sense similar to the use of parallel
>> universe (or superposition states) in a quantum computer. Also, the
>> comp
>> Platonia need not be greater that Sigma_1 Arithmetical truth
>> (which is
>> a tiny part of arithmetical truth, itself a tiny part of mathematical
>> truth): the deployment is really just the constructives
>> consequences of
>> 0, succession, addition and multiplication. And it is big as seen and
>> infered from inside, cf Hubble and ... the quantum multiverse. The
>> inaccessibility for manipulation is more of the type: no one can
>> make 17
>> even, not even a God.
>
> Agreed.
Best,
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Thu Jan 08 2009 - 13:42:44 PST